I think there is generally a misunderstanding as to why we use a fuse and how it changes according to types of appliance.
The fuse in the plug is a special case. And it has changed over the years. Years ago before we joined the EU if may have been permissible for manufacturers to rely on the fuse in the plug. But on joining the EU it was agreed that since many countries did not have fused plugs if the appliance needs a fuse under 16A then it must be included in the appliance and they should not rely on the one in the plug as in other countries there may not be a fuse in the plug. There is some slight variation some countries do have both 6A and 16A sockets, if fact at one time we had 5 and 15 amp sockets. So when fitting a British plug it may be required to have a 3A fuse (equivalent to using a 6A socket) or a 13A fuse. These are considered as preferred sizes.
Although the same physical size of fuse fits a fused connection unit (FCU) the rules change. With a fixed appliance i.e. one which is not plugged in, the manufacturer can stipulate the fuse size, so with a FCU you have 1, 2, 3, 5, 7, 10 and 13 amp fuses and the manufacturer can say what size should be fitted. He can also stipulate the type and I have needed to fit semi-conductor fuses to protect some items. Depending on the type and shape and location of an appliance it may be that fitting the fuse inside would cause problems. So often in the instructions it will say how it is to be supplied and the commissioning engineer has to insure the requirements are complied with. Be is special breakers or special fuses be it motor rated or semi-conductor.
So open up your TV and likely the fuse is built into the TV. But install a three phase compressor and likely the fuses are in a box on a near by wall.
With the fuse in the plug it is often hard to tell what size would be required. Some times you get a label 3A fuse installed but often nothing. So in the main as already said fuse is selected to match the cable used. Using a plug in energy meter it can be set to record the maximum power used. The results can be quite a surprise. My old fridge/freezer showed about 70W on run and stated 70W on the label however on start it showed 10A or to compare 2300 watt. And it states in the instructions never supply from an extension lead. However when replaced again run around 70W and the max showed as 120W which would be the size of the de-frost heater. Reason for the lower power peak is the use of an inverter drive.
Washing machine also has an inverter drive. However I wanted to run the washing machine from an inverter and 4 massive batteries and the boat engine and wood burning stove resulted in an abundant supply of hot water. Other than a special washing machine designed for boats seem to remember over £1500 we failed to find any washing machine with hot and cold fill. On enquiry we find something to do with energy labels plus if you heat items like egg they can be very hard to remove so they work better is washing is started with cold water then heated as the items are being washed. So only old washing machines had no water heater. All new washers have the heater built in. And when we hunted we could not find one under 2.2kW with an inverter motor which could be powered from a simulated sine wave.
So my dad taught me to read the appliance plate and if heating fit a fuse to just take that power, and if motor fit next size up. Which was the correct method in 1960 and still correct with a FCU. But with a plug look at size of cable under 1mm² fit a 3A otherwise fit a 13A. Nothing to do with perceived common sense, but more to do with being a member of the EU and all equipment should work in any member country simply by swapping the plug.